Reconstruction and interpolation of manifolds II: Inverse problems with partial data for distances observations and for the heat kernel

We consider how a closed Riemannian manifold M and its metric tensor g can be approximately reconstructed from local distance measurements. Moreover, we consider an inverse problem of determining $(M,g)$ from limited knowledge on the heat kernel. In the part 1 of the paper, we considered the approximate construction of a smooth manifold in the case when one is given the noisy distances d̃(x,y)=d(x,y)+εx,y for all points x,y∈ X, where X is a δ-dense subset of M and |εx,y|<δ. In this part 2 of the paper, we consider a similar problem with partial data, that is, the approximate construction of the manifold $(M,g)$ when we are given d̃(x,y) for x∈ X and y∈ U∩ X, where U is an open subset of M. In addition, we consider the inverse problem of determining the manifold $(M,g)$ with non-negative Ricci curvature from noisy observations of the heat kernel $G(y,z,t)$. We show that a manifold approximating $(M,g)$ can be determined in a stable way, when for some unknown source points zⱼ in X U, we are given the values of the heat kernel G(y,zₖ,t) for y∈ X∩ U and t∈ (0,1) with a multiplicative noise. We also give a uniqueness result for the inverse problem in the case when the data does not contain noise and consider applications in manifold learning. A novel feature of the inverse problem for the heat kernel is that the set M U containing the sources and the observation set U are disjoint.